A rating matrix R is the matrix (aspect-free) of m users and n items as in the classical recommender systems and R ( u , i ) = r u i represents the rating of user u on item i [16] . Generally, the ratings in a rating matrix are discrete numerical values.

A sentiment tensor S is a three-index tensor of m users, n items and k aspects. Here, S ( u , i , a ) = s u i a denotes the sentiment of user u about an item i along an aspect a. A value in S is either +1 or 0 or −1. 1 represents positive sentiment, −1 represents negative sentiment and 0 represents no sentiment.

There may be different ways to define the popularity and controversy of an aspect of an item. Simple definitions of popularity and controversy of an aspect of an item, computable using the Sentiment Tensor S are as explained in Sections 2.2.1 and 2.2.2.

The popularity of aspects of a specific item can have a significant role in building an aspect-driven recommender system but are rarely used [38] . The aspect-level popularity of an item can be combined with aspect-level preference of users to build a CF based RS [35] . For an item i recommend to a user u, the popularity score of the aspects of that item can be used as the criteria to recommend the top aspects of that item to the user u.

The relationship between two or more users can be assessed based on how they rate the aspects of items. For instance, users can be related based on sentiments towards all aspects of an item or based on their sentiments towards one aspect but considering all the items.

In general, positively biased users are the users tending to review every item or aspect under consideration positively. The Most Positive Users (MPU) about all aspects of an item i, noted as MPU(i) are users who review most or all the aspects of item i positively.

Two users are said to be the most disagreeing users if they tend to review every item or aspect under consideration with extreme opposite values in the reviewing scale. The Most Disagreeing Users on a specific aspect a considering all the items, noted as MDU(a) are the users who have reviewed aspect a with opposite sentiment to each other for most, if not all of the items.

The Nearest Neighbors to a user u based on an aspect a considering all the items is a set denoted as NN(a) are the users who think most like the user u toward aspect a considering all the items.

In general, clustering of objects is the process of grouping objects in a way that objects belonging to the same group are more similar to each other based on certain criteria than to the objects in other groups [39] . In our study, a cluster of users expressing similar sentiments to all aspects of item i, denoted as ClSent(U_i) is a group of users who have most similar sentiments to all aspects of item i. Similarly, a cluster of users emphasizing similar aspects of item i, denoted as ClEmp(U_i) consists of users who emphasize similar aspects of item i.

The preference level of users toward different aspects is an important part of aspect-based CF approaches [40] [41] [42] . Usually, the importance of an aspect to a user is inferred from reviews and are involved in aspect-based CF as one of the latent factors [18] [43] [44] [45] . Here, we present a simple approach to compute emphasis of a user towards an aspect based on the information stored in Sentiment tensor S.

The emphasis score of a user u towards an aspect a, e_ua can be defined as the ratio of times the aspect a is reviewed by user u over the total number of items reviewed by u. The value of e_ua lies in [0,1]. Mathematically, the emphasis score of a user u toward an aspect a can be computed as:

E ( u , a ) = e u a = 1 u ∑ i ∈ I U | s u i a | (5)

Item-Based User Disagreement, IBUD ( u , v | i ) between two users u and v is the dissimilarity score between them based on their aspect-sentiments towards all aspects of item i. Similarly, Item-Based User Agreement, IBUA ( u , v | i ) between two users u and v is the similarity score between them based on their aspect-sentiments towards all aspects of item i.

Aspect-Based User Disagreement, ABUD ( u , v | a ) between two users u and v is the dissimilarity score between them based on their sentiments toward aspect a considering all the items. Similarly, Aspect-Based User Agreement, ABUA ( u , v | a ) between two users u and v is the similarity score between them based on their sentiments toward an aspect a considering all the items.

In general, similarity between two data objects is a numerical measure to determine how alike they are [46] . And the value of similarity in general is in [0, 1]. And dissimilarity is a numerical measure to find different two data objects are. Dissimilarity or distance between two objects, not necessarily lie between [0, 1] until normalized.

The distance or similarity measure between objects is a key step in data mining tasks like classification and clustering [47] . The distances may be computed in different ways based on the type of data we are dealing with. Distance between two numerical or ordinal vectors x and y can generally be defined by any mathematical norm for the difference vector d ( x , y ) ∈ R n [48] . The Minkowski distance of different orders can be used to compute the distance between vectors formed from the numerical and ordinal data [49] . Minkowski distance between the ordinal vectors x and y of order p can be computed as:

d ( x , y ) = ( ∑ i = 1 n | x i − y i | p ) 1 p (6)

Minkowski distance of order 1 (p = 1) is Manhattan distance or 1-norm. And Minkowski distance of order 2 (p = 2) is the euclidean distance or 2-norm. K-means clustering which is one of the widely used unsupervised machine learning algorithms also uses the Minkowski distance of different order during clustering.

In this study, the distance between two users u and v based on item i, d ( u , v | i ) is termed as Item-Based User Disagreement (IBUD). IBUD ( u , v | i ) is computed based on Euclidean distance of aspect-sentiments of u and v based on all aspects in A of item i. Mathematically,

IBUD ( u , v | i ) = d ( u , v | i ) = ∑ a ∈ A ( s u i a − s v i a ) 2 (7)

IBUD may be normalized as required by the problems. For a normalized IBUD denoted as IBUD* which lies in [0,1], we define Item-based User Agreement (IBUA) as IBUA = 1 − IBUD*. In this work, the similarity between two users u and v based on an item i, sim ( u , v | i ) = IBUA ( u , v | i ) is computed based on the aspect sentiments of users u and v towards all aspect in A ′ of item i. And the weight between two users u and v is defined as

w u v = ∑ i ∈ I u ∩ I v ( u , v | i ) (8)

The distance between two users u and v based on aspect a, d ( u , v | a ) is termed as Aspect-Based User Disagreement (ABUD). ABUD ( u , v | a ) can be computed considering all the items in I as:

ABUD ( u , v | a ) = d ( u , v | a ) = ∑ i ∈ I ( s u i a − s v i a ) 2 (9)

ABUD may be normalized as required by the problems. For a normalized ABUD denoted as ABUD* which lies in [0, 1], we define Aspect-based User Agreement (ABUA) as ABUA = 1 − ABUD*.

The algorithm for finding the most controversial aspect a of an item i noted as ConAsp(i) is as below:

1) For each aspect a of item i,

a) Compute popularity l a i + using Equation (1) and its compliment l a i − using Equation (2).

b) Compute controversy κ a i of the aspect a of the item i using Equation (4).

2) The most controversial aspect a of item: ConAsp(i) = aspect with the maximum κ a i .

The idea of finding the most controversial aspect may look like a simpler problem compared to the big machine learning problems in the Recommendation Systems. But the solution of this problem can play a vital role in making meaningful recommendations to the users when combined with other solutions.

The example in Table 2 gives the controversy κ a i of aspects of the item 0. This example is based on the Hotel dataset used in this study. The aspect Value is the most controversial aspect of hotel 0, because it has the highest controversial measure κ a i among the six aspects of the hotels under consideration.

The algorithm for finding the most controversial item i based on an aspect a noted as ConItem(a) is as below:

1) For each item i,

a) Compute popularity l a i + using Equation (1) and its compliment l a i − using Equation (2) for specific aspect a.

b) Compute controversy κ a i of the specific aspect a of item i using Equation (4) and store it.

2) The most controversial item i based on specific aspect a: ConItem(a) = item with the maximum κ a i for specific aspect a.

The solution to the problem of finding the most controversial item based on an aspect a can also be a solution to the challenge of recommending items to new users. The most controversial items can be avoided while recommending to new users with insufficient rating or sentiment data.

For the hotel data used in this study, item 300 is the most controversial item based on aspect Location. The result is as disclosed in Table 3.

Let U_i be the set of all users that have reviewed the item i. The proposed algorithm

to solve the problem of finding the top N users who are most positive about all aspects of an item i as below:

1) Assume a reference user u’ as a user who has got positive sentiments for all the aspects of the item i.

2) Compute the dissimilarity, Item-Based User Disagreement IBUD ( u , v | i ) of each user in U_i with the assumed user u’ using Equation (7).

3) Sort the users in U_i based on the dissimilarity values IBUD ( u ′ , u | i ) ascendingly.

4) MPU ( A | i ) = top N sorted users of U_i. Here, MPU ( A | i ) are the N users that are the N most positive about all aspects of the item i.

Here, in Table 4, we give an example solution of top 5 most positive users about all aspects of item 7. This example is based on Hotel Dataset.

Based on the proposed solution for finding the top N users who are most positive about all aspect of item i, a hypothesis can be proposed as:

Hypothesis1: The top N users who are most positive about k − 1 aspects of an item are likely to positive about the k^th aspect, which has not been used for finding the top N most positive users.

We evaluated Hypothesis1 by introducing an approach called Leave One Aspect Out. The steps involved during this evaluation are as follows:

I = set of all items,

A = set of all aspects.

1) For an item i in I,

a) U_i = set of all users that have reviewed item i

b) For an aspect a in A,

i) Split the data into training and test data,

Let, A ′ = A − a ,

Trn = sub-tensor S ( u , i , a ′ ) for all u ∈ U i , a ′ ∈ A ′ is training data.

Tst = sub-tensor S ( u , i , a ) for all u ∈ U i is testing data.

ii) Let MPU = set of N most positive users about all aspects in A’ found using approach in Section 4.3.

iii) Find n a i + | MPU = number of users with positive sentiment towards aspect a of item i for all u ∈ MPU .

iv) Find n a i − | MPU = number of users with negative sentiment towards

aspect a of item i for all u ∈ MPU .

v) Accuracy [51] = n a i + | MPU n a i + | MPU + n a i − | MPU , accuracy obtained is store in a list accL.

c) Step 1b is performed for each aspect in A.

2) Step 1 is performed for each item in I

3) Overall accuracy measure = arithmetic means of accuracy values in accL of all aspects and items

A sentiment s u i a of a user u ∈ MPU for aspect a in the test set is not considered during evaluation if sentiment s u i a = 0 .

The results of Evaluation of Hypotheis 1 using the Hotel Dataset are tabulated in Table 5.

The algorithm to find top N users who feel most like a specified user u’ based on an aspect a is as follows:

1) Find I ′ u = the set of all items reviewed by user u'

2) Find U' = the set all users reviewing at least one item in I u ′

3) For all u ∈ U ′ , compute Aspect-Based User Disagreement ABUD ( u ′ , u | a ) between users u’ and u based on sentiment toward aspect a considering items rated both and normalize by common number of items reviewed by users u’ and u. Normalized ABUD ( u ′ , u | a ) can be computed as:

ABUD * ( u ′ , u | a ) = 1 | I u ∩ I u ′ | ∑ i ∈ I u ∩ I u ′ ( s u ′ i a − s u i a ) 2 (10)

where I u is the set of items reviewed by u,

I u ′ is the set of items reviewed by u’ and,

| I u ∩ I u ′ | is the cardinal number of set of items reviewed by both I u and I ′ u .

4) Sort ABUD * ( u ′ , u | a ) ascendingly and the user u associated with ABUD * ( u ′ , u | a ) .

5) N nearest neighbors to u' based on aspect a, NN ( u ′ | a ) = top N ascendingly sorted users based on ABUD * ( u ′ , u | a ) . Hence the top N sorted users

based on the values of ABUD * ( u ′ , u | a ) are the users who think most like u’ based on an aspect a.

Table 6 gives an example solution for finding the top 5 users who feel most like the user 10 based on the aspect Rooms. This example is based on Hotel Dataset.

Let U = set of all the users and I = set of all the items. Then, the top N pair of users disagreeing most on a specific aspect a considering all the items noted as MDU ( a | I ) can be found using the following steps:

1) Compute Aspect-Based User Disagreement ABUD ( u ′ , u | a ) between users u’ and u for u ∈ U , u ′ ∈ U , u ≠ u ′ considering the sentiments of a specific aspect a of all the items in I using Equation (9). In other words, compute distance between each pair of users in U considering sentiments of the specific aspect a of all items in I.

2) Sort pair of users ( u , u ′ ) descendingly based on distances ABUD ( u ′ , u | a ) . MDU ( a | I ) = top N descendingly sorted ( u , u ′ ) based on ABUD ( u ′ , u | a ) . Hence, MDU ( a | I ) are the pair of users who disagree most on the considered specific aspect a considering all the items in I.

Table 7 gives an example solution for finding the top 5 pairs of users who

disagree most on aspect Location considering all the items based on the Hotel dataset.

To find the groups of users who are most similar in all aspects of an item i, we can cluster the users based on the sentiment values users provided to all aspects of item i. The K-means clustering algorithm for finding the groups of users who are most similar in all aspects of an item i is used as follows.

1) Find U_i = the set of users that have reviewed item i.

2) Cluster the users in U_i based on their sentiments toward all aspects of item i using K-means clustering.

This algorithm uses Item-Based User Disagreement IBUD ( u , u ′ | i ) computed using Equation (7) during K-means clustering.

Clusters obtained are the groups of users who are most similar in all the aspects of an item i.

Figure 1 provides an example solution of finding the groups of users who are most similar in all aspects of an item i. This example is based on Hotel Dataset. One can see 5 clusters or groups of users who are most similar in all aspects of item 99.

The group of users who emphasize the same aspects of an item i can be found by clustering the users reviewing the same aspects of item i. In our approach, we cluster the users by treating positive and negative sentiment as same. We use K-means clustering algorithm to find users who emphasize same aspect of item i as follows:

1) Find U_i = the set of the users that have reviewed the item i. Then,

2) Cluster the users in U_i based on their sentiments toward all aspects of the item i, but by treating positive and negative sentiment as same. Here, the K-means clustering is performed using distance IBUD ( u , u ′ | i ) based on the absolute value of the aspect sentiments. Equation (7) is modified as below to compute modified IBUD ( u , u ′ | i ) .

IBUD ( u , u ′ | i ) = ∑ a ∈ A ( | s u i a | − | s u ′ i a | ) 2 (11)

Figure 2 provides an example solution of finding the groups of users who emphasize the same aspects of item 99 in the Hotel Dataset.

The aspects of an item can be ranked based on the emphasis given by a user u to them using following steps:

1) Find I_u = set of items reviewed by user u and A = set of all aspects.

2) Compute the emphasis score of the user u toward each aspect a using Equation (5).

3) Sort E ( u , a ) by aspects descendingly and rank. The aspect with the highest value of E ( u , a ) will be the most emphasized aspect.

Table 8 presents an example solution for ranking the aspects based on emphasis given by user 10 to them. This example is based on the Hotel dataset. Table 7 shows the emphasis score of user 10 towards each aspect in A. and indicates that user 10 gives strong emphasis to the aspects Service and Cleanliness whereas aspect Sleep Quality is of least emphasis to user 10.

Here, we presented eight aspect-based CF tools in ABCFT as a start of compiling the tools that can be extracted from Aspect-Sentiment Tensor. And, we would like to encourage exploring and adding the new tools to ABCFT, so the area of recommendation techniques using aspect-based information can grow rapidly.